In a triangle, two of the side lengths are 7 and 8, and the angle between them is $120^\circ.$  Find the length of the third side.
By the Law of Cosines, the third side is
\[\sqrt{7^2 +  8^2 - 2 \cdot 7 \cdot 8 \cos 120^\circ} = \sqrt{7^2 + 8^2 + 7 \cdot 8} = \boxed{13}.\]